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Group Algebra of a Discrete Group and Different Notions of a Group Algebra?

0

$begingroup$

I am reading these two wiki articles: https://en.wikipedia.org/wiki/Group_algebra and https://en.wikipedia.org/wiki/Pontryagin_duality

From my understanding, the group algebra of a topological group G is the collection of all continuous functions $f : G to BbbC$ with compact support, denoted by $C_c(G)$. When $G$ is discrete, compact support implies finite support, so we can identify $f : G to BbbC$ with the sum $sum_g in G f(g) g$, which shows that the group ring and group algebra are isomorphic for discrete groups. Does this sound right?

However, I am confused by a passage in the second link given above:

This algebra [$L^1(G)$] is referred to as the Group Algebra of $G$

What is in fact the group algebra?

continuity topological-groups duality-theorems

share|cite|improve this question

edited Apr 2 at 2:03

Saad

20.7k92452

asked Mar 31 at 18:56

user193319user193319

2,4622928

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A group algebra is what you said, see wikipedia. For discrete groups, it is the group ring, also referred to as group algebra.
  $endgroup$
  – Dietrich Burde
  Mar 31 at 19:00
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DietrichBurde So, most people/resources don't refer to $L^1(G)$ as the Group Algebra of $G$?
  $endgroup$
  – user193319
  Mar 31 at 19:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It may depend on the context, but usually "group algebra" is - well, see here.
  $endgroup$
  – Dietrich Burde
  Mar 31 at 19:07
  
  

  
  
  
  

add a comment | 

0

$begingroup$

I am reading these two wiki articles: https://en.wikipedia.org/wiki/Group_algebra and https://en.wikipedia.org/wiki/Pontryagin_duality

From my understanding, the group algebra of a topological group G is the collection of all continuous functions $f : G to BbbC$ with compact support, denoted by $C_c(G)$. When $G$ is discrete, compact support implies finite support, so we can identify $f : G to BbbC$ with the sum $sum_g in G f(g) g$, which shows that the group ring and group algebra are isomorphic for discrete groups. Does this sound right?

However, I am confused by a passage in the second link given above:

This algebra [$L^1(G)$] is referred to as the Group Algebra of $G$

What is in fact the group algebra?

continuity topological-groups duality-theorems

share|cite|improve this question

edited Apr 2 at 2:03

Saad

20.7k92452

asked Mar 31 at 18:56

user193319user193319

2,4622928

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A group algebra is what you said, see wikipedia. For discrete groups, it is the group ring, also referred to as group algebra.
  $endgroup$
  – Dietrich Burde
  Mar 31 at 19:00
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DietrichBurde So, most people/resources don't refer to $L^1(G)$ as the Group Algebra of $G$?
  $endgroup$
  – user193319
  Mar 31 at 19:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It may depend on the context, but usually "group algebra" is - well, see here.
  $endgroup$
  – Dietrich Burde
  Mar 31 at 19:07
  
  

  
  
  
  

add a comment | 

$begingroup$

I am reading these two wiki articles: https://en.wikipedia.org/wiki/Group_algebra and https://en.wikipedia.org/wiki/Pontryagin_duality

From my understanding, the group algebra of a topological group G is the collection of all continuous functions $f : G to BbbC$ with compact support, denoted by $C_c(G)$. When $G$ is discrete, compact support implies finite support, so we can identify $f : G to BbbC$ with the sum $sum_g in G f(g) g$, which shows that the group ring and group algebra are isomorphic for discrete groups. Does this sound right?

However, I am confused by a passage in the second link given above:

This algebra [$L^1(G)$] is referred to as the Group Algebra of $G$

What is in fact the group algebra?

continuity topological-groups duality-theorems

share|cite|improve this question

edited Apr 2 at 2:03

Saad

20.7k92452

asked Mar 31 at 18:56

user193319user193319

2,4622928

$endgroup$

share|cite|improve this question

edited Apr 2 at 2:03

Saad

20.7k92452

asked Mar 31 at 18:56

user193319user193319

2,4622928

share|cite|improve this question

edited Apr 2 at 2:03

Saad

20.7k92452

asked Mar 31 at 18:56

user193319user193319

2,4622928

edited Apr 2 at 2:03

Saad

20.7k92452

edited Apr 2 at 2:03

Saad

20.7k92452

asked Mar 31 at 18:56

user193319user193319

2,4622928

asked Mar 31 at 18:56

user193319user193319

2,4622928