A Family of Semi-norms in C^*-algebras

We introduce a new family of non-negative real-valued functions on a C^*-algebra A, i.e., for 0≤ μ ≤ 1, \|a\|_σμ= sup √|f(a)|² σμ f(a^*a): f∈ A', \, f(1)=\|f\|=1 , where a∈ A and σμ is an interpolation path of the symmetric mean σ. These functions are semi-norms as they satisfy the norm axioms, except for the triangle inequality. Special cases satisfying triangle inequality, and a complete equality characterization is also discussed. Various bounds and relationships will be established for this new family, with a connection to the existing literature in the algebra of all bounded linear operators on a Hilbert space.