Lip-norm On Subalgebra And Stability Of Haagerup-type Condition By Adopting Direct Sum And Tensor Product

Let A be an unital pre-C*-algebra.A Lip-norm on A is a seminorm with some conditions,that is the generalization of Lipschitz seminorm of ordinary compact metric spaces.The paper proves that if A has a Lip-norm,then any unital *-subalgebra of A has a Lip-norm by setting up a linear mapping,this is a generalization of Rieffel' conclusion. In[2],Ozawa and Rieffel show that if A has a *-filtration and satisfies Haagerup-type condition,then we can define a Lip-norm on A by Dirac operator.In[2],Ozawa and Rieffel also show that the above conclusion is stable by adopting reduced free product.The paper proves:the above conclusion is stable by adopting finite direct sum,Haagerup-type condition applies to the tensor product C*-algebra consisting of an unital pre-C*-algebra satisfing Haagerup-type condition and a finite dimension C*-algebra.