Existence of Bounded Approximate Identities for Modules of a Normed Algebra
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In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given:
Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)a oversetlambda to a $ and $ ae(lambda) oversetlambda to a $ for each $ a in A $.
Definition Let $ A $ be a normed algebra and let $ X $ be a normed $ A $-module. A Bounded Approximate Identity in $ A $ for $ X $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)x oversetlambda to x $ and $ xe(lambda) oversetlambda to x $ for each $ x in X $.
My question is, if $ A $ is a normed algebra that has a bounded approximate identity, then must it have a bounded approximate identity for each of its normed $ A $-modules?
Is it true if $ A $ is a Banach algebra and $ X $ is a Banach $ A $-module?
banach-algebras
asked Aug 19 at 3:32
LMW
987
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up vote
0
down vote
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In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given:
Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)a oversetlambda to a $ and $ ae(lambda) oversetlambda to a $ for each $ a in A $.
Definition Let $ A $ be a normed algebra and let $ X $ be a normed $ A $-module. A Bounded Approximate Identity in $ A $ for $ X $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)x oversetlambda to x $ and $ xe(lambda) oversetlambda to x $ for each $ x in X $.
My question is, if $ A $ is a normed algebra that has a bounded approximate identity, then must it have a bounded approximate identity for each of its normed $ A $-modules?
Is it true if $ A $ is a Banach algebra and $ X $ is a Banach $ A $-module?
banach-algebras
asked Aug 19 at 3:32
LMW
987
Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided...
– zipirovich
Aug 19 at 4:12
Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong?
– LMW
Aug 19 at 5:39
Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides.
– zipirovich
Aug 19 at 5:43
Hm, I thought the terms "module" and "bimodule" were synonymous.
– LMW
Aug 19 at 6:04
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given:
Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)a oversetlambda to a $ and $ ae(lambda) oversetlambda to a $ for each $ a in A $.
Definition Let $ A $ be a normed algebra and let $ X $ be a normed $ A $-module. A Bounded Approximate Identity in $ A $ for $ X $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)x oversetlambda to x $ and $ xe(lambda) oversetlambda to x $ for each $ x in X $.
My question is, if $ A $ is a normed algebra that has a bounded approximate identity, then must it have a bounded approximate identity for each of its normed $ A $-modules?
Is it true if $ A $ is a Banach algebra and $ X $ is a Banach $ A $-module?
banach-algebras
asked Aug 19 at 3:32
LMW
987
In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given:
Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)a oversetlambda to a $ and $ ae(lambda) oversetlambda to a $ for each $ a in A $.
Definition Let $ A $ be a normed algebra and let $ X $ be a normed $ A $-module. A Bounded Approximate Identity in $ A $ for $ X $ is a bounded net $ e(lambda) $ in $ A $ such that $ e(lambda)x oversetlambda to x $ and $ xe(lambda) oversetlambda to x $ for each $ x in X $.
My question is, if $ A $ is a normed algebra that has a bounded approximate identity, then must it have a bounded approximate identity for each of its normed $ A $-modules?
Is it true if $ A $ is a Banach algebra and $ X $ is a Banach $ A $-module?
banach-algebras
asked Aug 19 at 3:32
LMW
987
asked Aug 19 at 3:32
LMW
987
asked Aug 19 at 3:32
LMW
987
asked Aug 19 at 3:32
LMW
987
987
Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided...
– zipirovich
Aug 19 at 4:12
Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong?
– LMW
Aug 19 at 5:39
Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides.
– zipirovich
Aug 19 at 5:43
Hm, I thought the terms "module" and "bimodule" were synonymous.
– LMW
Aug 19 at 6:04
add a comment |Â
Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided...
– zipirovich
Aug 19 at 4:12
Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong?
– LMW
Aug 19 at 5:39
Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides.
– zipirovich
Aug 19 at 5:43
Hm, I thought the terms "module" and "bimodule" were synonymous.
– LMW
Aug 19 at 6:04
Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided...
– zipirovich
Aug 19 at 4:12
Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided...
– zipirovich
Aug 19 at 4:12
Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong?
– LMW
Aug 19 at 5:39
Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong?
– LMW
Aug 19 at 5:39
Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides.
– zipirovich
Aug 19 at 5:43
Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides.
– zipirovich
Aug 19 at 5:43
Hm, I thought the terms "module" and "bimodule" were synonymous.
– LMW
Aug 19 at 6:04
Hm, I thought the terms "module" and "bimodule" were synonymous.
– LMW
Aug 19 at 6:04
add a comment |Â
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Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided...
– zipirovich
Aug 19 at 4:12
Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong?
– LMW
Aug 19 at 5:39
Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides.
– zipirovich
Aug 19 at 5:43
Hm, I thought the terms "module" and "bimodule" were synonymous.
– LMW
Aug 19 at 6:04